Question: A large brine tank containing a solution of salt and water is being diluted with fresh water. The relationship between the elapsed time, $t$, in hours, after the dilution begins, and the concentration of salt in the tank, $S(t)$, in grams per liter $(\text{g/l})$, is modeled by the following function. $ S(t)=500\cdot e^{{-0.25t}}$ What will the concentration of salt be after $10$ hours? Round your answer, if necessary, to the nearest hundredth.
Solution: Thinking about the problem We want to find the concentration of salt in the tank after $10$ hours. In other words, we are given a $t$ value of $10$ hours and want to find the salt concentration associated with that input, or $S(10)$. To do this, we can substitute ${10}$ in for $ t$ and evaluate. $ S( {10})=500\cdot e^{{-0.25({10})}}$ Evaluating the expression We can use a calculator to evaluate the expression. The answer is shown below. $\begin{aligned}S(10)&=500\cdot e^{{-0.25(10)}}\\\\ &=500\cdot e^{{-2.5}}\\\\ &\approx41.04\\\\ \end{aligned}$ After $10$ hours, the concentration of salt will be $41.04$ $\text{ g/l}$.